GATE CE 2015 SET-2

 Question 1
While minimizing the function $f(x)$, necessary and sufficient conditions for a point, $x_{0}$ to be a minima are:
 A ${f}'(x_{0}) \gt 0\: and\: \: {f}''(x_{0})=0$ B ${f}'(x_{0}) \lt 0\: and\: \: {f}''(x_{0})=0$ C ${f}'(x_{0})=0\: and\: \: {f}''(x_{0}) \lt 0$ D ${f}'(x_{0})=0\: and\: \: {f}''(x_{0}) \gt 0$
Engineering Mathematics   Calculus
Question 1 Explanation:
$f(x)$ has a local minimum at $x=x_{0}$
if ${f}'\left ( x_{0} \right )=0$
and ${f}''\left ( x_{0} \right ) \gt 0$
 Question 2
In Newton-Raphson iterative method, the initial guess value ( $x_{ini}$) is considered as zero while finding the roots of the euation:$f(x)=-2+6x-4x^{2}+0.5x^{3}$. The correction, $\Delta x$, to be added to $x_{ini}$ in the first iteration is____________.
 A 0.5 B 0.33 C 0.8 D 0.2
Engineering Mathematics   Numerical Methods
Question 2 Explanation:
\begin{aligned} f\left ( x \right )&=-2+6x-4x^{2}+0.5x^{3} \\ {f}'\left ( x \right )&=6-8x+1.5x^{2} \\ x_{ini}&=0\\ \text{By Newton }&\text{Raphson Method,}\\ x_{1}&=x_{ini}-\frac{f\left ( x_{ini} \right )}{{f}'\left ( x_{ini} \right )} \\ &= 0-\frac{-2}{6} \\ \Rightarrow \;\;x_{1}&=\frac{1}{3} \\ \therefore \;\; \Delta x&=x_{1}-x_{ini}=\frac{1}{3} \end{aligned}

 Question 3
Given , $i=\sqrt{-1}$, the value of the definite integral, $I=\int_{0}^{\pi/2}\frac{\cos x+i\sin x}{\cos x-i\sin x}dx$ is:
 A 1 B -1 C i D -i
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} I&=\int_{0}^{\pi /2}\frac{\cos x+i\sin x}{\cos x-i\sin x}dx \\ &=\int_{0}^{\pi /2}\frac{e^{ix}}{e^{-ix}}dx=e^{2ix}dx \\ &=\frac{1}{2i}\left [ e^{2ix} \right ]_{0}^{\pi /2} \\ &=\frac{1}{2i}\left [ e^{i\pi }-1 \right ] \\ &=\frac{1}{2i}\left ( -1-1 \right ) \\ &=-\frac{1}{i}=i \end{aligned}
 Question 4
$\lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^{2x}$ is equal to
 A $e^{-2}$ B e C 1 D $e^{2}$
Engineering Mathematics   Calculus
Question 4 Explanation:
$y=\: \lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^{2x}$
$\log y = \lim_{x\rightarrow \infty }2x\log \left ( 1+\frac{1}{x} \right )$
Which is in the form of $\infty \times 0$.
To convert this into $\frac{0}{0}$ form, we rewrite as,
$\Rightarrow \log y= \lim_{x\rightarrow \infty }\frac{2\log \left ( 1+\frac{1}{x} \right )}{1/x}$
Now is in $\frac{0}{0}$ form.
Using L'Hospital's Rule,
\begin{aligned} \log y&=\lim_{x\rightarrow \infty }\frac{\frac{2\times -\frac{1}{x^{2}}}{1+\frac{1}{x}}}{-\frac{1}{x^{2}}} \\ \log y&=\lim_{x\rightarrow \infty }\frac{2}{1+\frac{1}{x}}=2 \\ \therefore \;\; y&= e^{2}\end{aligned}
 Question 5
Let $\mathbf{A}=\left [ a_{ij} \right ],\; \; \; 1\leq i, \; j\leq n$ with $n\geq 3\; and\; a_{ij}=i.j$. The rank of A is:
 A 0 B 1 C n-1 D n
Engineering Mathematics   Linear Algebra
Question 5 Explanation:
Rank of A=1
Because each row will be scalar multiple of first row.So we will get only one-zero row in row Echeleaon form of A.

Alternative:
Rank of A=1
Because all the minors of order greater than 1 will be zero.

There are 5 questions to complete.